(diluted 5-fold with p H 1.7 buffer containing HgCL a t 50 “C) would produce a n absorbance (A,,, = 0.06 after 48 minutes) which could be measured accurately using the Beckman-Gilford spectrophotometer. Since benzoylation of the side chain amino group and subsequent rearrangement of a-benzamidobenzylpenicillin to form a-benzamidobenzylpenicillenic acid is specific for intact ampicillin (6),the overall procedure could be employed for the spectrophotometric assay of ampicillin both in kinetic reaction solutions and presumably in solutions containing a mixture of ampicillin and another penicillin. For example, treatment of an aliquot of an aqueous solution of benzylpenicillin and ampicillin at pH 2.1 in water at ambient temperature would yield benzylpenicillenic acid only after ca. 30 minutes (3) (measured spectrophotometrically at 322 nm) since the rate of degradation of ampicillin under such condi-
tions is negligible (9). Benzoylation of another aliquot of the mixture and treatment of an aliquot of the resultant solution a t p H 1.7 and 50 O C in 19% ethanol-water would yield both benzylpenicillenic acid and a-benzamidobenzylpenicillenic acid (from ampicillin). However, an absorbance measured a t 322 nm after 48 minutes would correspond almost entirely to a-benzamidobenzylpenicillenic acid since the benzylpenicillenic would have decomposed completely under these conditions. ACKNOWLEDGMEhT
The authors thank Bristol Laboratories for the gift of sodium ampicillin.
RECEIVED for review August 3, 1970. Accepted December 2, 1970.
Application of a Computerized Electrochemical System to Pulse Polarography at a Hanging Mercury Drop Electrode H. E. Keller’ and R. A. Osteryoung Department of‘ Chemistry, Colorado State Unicersity, Fort Collins, Colo. 80521
Application of computerized pulse polarography on a hanging drop to analysis of extremely dilute solutions i s demonstrated. An approximate theory is developed which shows that for reversible systems functionally identical behavior can be expected on the dropping and hanging drop mercury electrodes. A decrease in sensitivity for irreversible reactions would be observed under otherwise identical conditions with the stationary electrode. Ensemble averaging and digital smoothing are described and their effect on signalto-noise ratio is demonstrated. Variations of pulse height, pulse width, and time between pulses are briefly discussed. Response obtained on 4 X 10-8M Cd2+solution indicates that usable data can be obtained at this level while a precision of 10% i s indicated on 4 x lO-’M Cd2+.
DERIVATIVE MODE PULSE POLAROGRAPHY has been shown to be avery sensitive analytical technique (Z-4). When performed on a stationary electrode, additional advantages may accrue such as increased electrode area (5), increased speed of analysis, and ensemble-averaging (6, 7) undisturbed by drop area uncertainty. Computerization of chemical analysis is becoming very popular today, a fact occasioned by utility and by novelty. In electrochemical analysis, several workers have been enPresent address, Department of Chemistry, Northeastern University. Boston, Mass. 02115 (1) E. Temmerman and F. Verbeek, J . Electroanal. Chem., 12, 158 (1966). ( 2 ) A. Lagrou and F. Verbeek, ibid., 19, 413 (1968). (3) E. P. Parry and R. A. Osteryoung, ANAL.CHEM., 36, 1366 (1964). (4) C. Peker, M. Herlem, and J. Badoz-Lambling, Freseiiius’ Z. Anal. Chem., 224,204 (1967). ( 5 ) G. D. Christian, J . Electroanal. Chem., 22, 333 (1969). (6) S. P. Perone, J. E. Harrar, F. B. Stevens, and R. E. Anderson, ANAL.CHEM., 40, 899 (1968). (7) V. W. Lee, T. P. Cheatham, Jr., and J. B. Wiesner, Proc. I.R.E., 38 1165 (1950).
342
ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
gaged in demonstrating the utility of an on-line computer system (6, 8-10). By employing a computer to take measurements, control the experiment, and analyze the resulting data, maximum use is made of the advantages of derivative pulse polarography at a stationary electrode. Other capabilities such as convolution of the current response to increase the signal-to-noise ratio and automatic determination of peak positions and heights by a real-time successive approximation technique can be developed readily on a computer system. In this paper the characteristics of a computerized electrochemical system are described and demonstrated in an application to pulse polarography a t a stationary electrode. Some of the advantages of the computer system over conventional systems are developed; some further potential advantages are mentioned. The realizable sensitivity of the system as an analytical tool seems, at present, to be limited by background. Instrumental artifacts and oxygen appear to be the primary contributors. With the problems, however, measurable response is obtained with 4 x 10-SM Cd2+. Reducing the concentration of supporting electrolyte to below 10d3Mis an important factor in this achievement. This sensitivity compares with stripping analysis where sensitivities as low as 6 X 10-11M (ZZ) have been reported. More normally, values of 10-9M are seen with respect to this technique. It does require that the species determined be concentrated into another phase, a fact which limits its general utility. Potential sweep voltammetry has a reported sensitivity of -~10-~M( 1 2 ) (8) G. Lauer and R. A. Osteryoung. ANAL.CHEM.,40 (IO), 30A (1968). (9) G. P. Hicks, A. A. Eggert, and E. C. Toren, Jr., ibid., 42, 729 (1970). (10) G. Lauer, R. Abel, and F. C. Anson, ibid., 39,765 (1967). (11) S. P. Perone and J. R. Birk, ibid., 37, 9 (1965). (12) J. W. Ross, R. D. DeMars, and I. Shain, ibid.. 28, 1768 (1956).
REAL-TIME CLOCK
1/o
PDP- 8/I CPU a CORE (4K)
DEVICE SELECT-
I
CONVERTER
LINES B,
X-Y
PLOTTER
SKIP
Figure 1. System block diagram
trode was a platinum wire separated from the solution by a pinhole in the end of a piece of glass tubing. All deaerations were performed for at least 15 minutes. A glass tube with a small hole in the end was used as the nitrogen inlet.
and can be extended by an order of magnitude by analog differentiation (13). For a discussion of these and other electroanalytical techniques, the reader is referred to reference (14). EXPERIMENTAL
SYSTEM DESCRIPTION
A PDP-8/1 computer (Digital Equipment Corp.) was used for all computing, control, and measurement functions except for the use of a potentiostat and current amplifier, which employed Philbrick-Nexus SP656 and P65AU operational amplifiers. A more complete description of the digital system is given in the next section. All chemicals were reagent grade and used without further purification. The water was twice-distilled, the first distillation being from alkaline permanganate solution. A spoutless 100-ml beaker was used as the electrochemical cell. A commercial cover (Beckman Instruments) was adapted to use in this experiment, and prepurified nitrogen was always kept flowing over or through the solution. No frits were used in the apparatus. The indicating electrode was a Brinkmann microburet hanging mercury drop electrode. The capillary was dewetted with dichlorodimethylsilane prior to use and the end broken off. The reference was a Sargent SCE. The auxiliary elec-
The system, Figure 1, consists of a PDP-8/1 computer interfaced with a real-time clock, and X-Y plotter, and a potentiostat plus current follower system based on operational amplifiers. All operations are under computer control. The real time clock can be set under program control to turn on a flag after the passage of from 1 to 4096 clock pulses or “ticks.” This flag is connected to computer skip line so that it may be interrogated by an I/O command. Normally the program uses the skip instruction in a wait loop. The clock can also be read at any time to determine the elapsed time, and it will provide a computer interrupt when the flag is set if the interrupt is enabled. For a more complete description of the clock, see reference (15). Further information on the interface is contained in the article by Lauer and Osteryoung (8). The plotter is a Hewlett-Packard X-Y plotter with a digital plotting accessory. The two axes are driven by 10-bit D/A converters. Completion of the plotting of a point is signalled to the computer from the plotter.
(13) S. P. Perone and T. M. Mueller, Anal. Chern., 37, 2 (1965). (14) H. A. Laitinen, in “Trace Characterization, Chemical and Physical,” W. W. Meinke and B. F. Scribner, Ed., Nar. Bur. Stand. US.Monogr. 100, 1967.
a %2 EAD PARAWTERS
INITIALIZE
PARAMETERS V e r t , Amp,
o l t . Increment SET
Resistance
WAIT, DELAY
DIVIDE SUM BY NO. OF PULSES
ype of-Exp.
urrent Meas.
~
(15) D. M. Mohilner and P. R. Mohilner, O.N.R. Tech. Report No. 1, Project NR359-493 (July 1969).
FIRST
p l s e Ht,
(D Mode Only) e l a y Time
UTPUT E+hE
k l s e Width (AT k i c k Mark S e p . ‘1 L
I
I
WAIT8 AT
Figure 2. Program block diagram
& IC1 - I
ALLOW USER TO DETERMINE PARAMETERS TO CHANGE
co
+No
OMPLET
4NALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
343
The electrochemical circuitry is conventional. Input to the potentiostat comes from a variable voltage source and from a 12-bit D/A converter interfaced with the computer. This converter, when properly connected, has a resolution of 0.5 mV for a 2-volt scan. It is run with a single digital buffer into which information is input from the computer by a jamtransfer method. This method of transfer eliminates the need to clear the buffer before it can receive an input and, therefore, prevents spurious signals during transfer. Switching time of the converter is about 3 psec. The current output from the cell is converted to a voltage through a current follower. The output from this follower is input to a 12-bit A/D converter. This converter has a sample and hold accessory which is activated by program command. The aperture time is 150 nsec while the track time is 12 psec. The converter begins conversion on command and gives a conversion complete signal. Conversion time is 35 psec. The conversion complete signal may also produce an interrupt if activated.
For plotting, provision is made to input horizontal and vertical tick mark spacings. During the experiment, the current is measured before and at the end of each pulse voltage which is applied to the electrode. After completion of the requisite number of averaging cycles (pulses) at one potential, the calculated point is plotted and the appropriate digital representation of the potential is incremented. When the experiment is completed, the stored results may be printed. Also, a digitally smoothed plot may be obtained. If a repeat is desired, it can be made with a change of the plotter vertical amplification factor (a program calculation), the plotter bias (also programmed), and/or the number of averaging cycles simply by answering “Y” (Yes) to the computer’s query. For other alterations, the answer must be “N” (No). In the latter case the computer switches are set to indicate the changes desired. The computer will ask for only the parameters selected.
PROGRAM DESCRIPTION
THEORY
A block diagram of the program is given in Figure 2. A description of the terms used and certain processes not obvious in the diagram follows. The experimental parameters are read in a questionanswer mode. The clock cycle, voltage division factor, and current measuring resistor values allow time, voltage, and current parameters to be communicated in engineering units rather than in computer internal representation. The voltage division factor is an attenuation between the appropriate D/A converter and the potentiostat. The full 12-bit precision of the D/A converter is realized only if its full scale output of 10 volts can be utilized. Therefore the 10-volt output is divided to result in a control voltage slightly greater than the desired scan range. This division is accomplished by utilizing the fact that the input to the potentiostat is a summing network. Appropriate choice of summing resistor for the D/A input results in the desired scaling. There are three types of pulse polarography that may be performed. The first, designated N (for normal or integral mode), is produced by maintaining the potential between pulses at a constant value. Each successive pulse is of greater magnitude than the preceding one, and the current response at a fixed time after the application of a pulse is plotted against pulse height. In the D (for derivative) mode, all pulses are of equal height. However, the potential between pulses is stepped to a greater value after each pulse. The current is plotted against the potential prior to application of the pulse. A third mode, Q, has been incorporated to allow the difference between successive pairs of current responses output in the N mode to be plotted. This mode is termed the difference mode. Although the D and Q modes give functionally identical behavior when used on reversible systems, the fact that the waiting period between pulses is spent at different potentials can produce marked differences in behavior for systems with complications. An example might be a system exhibiting absorption where desorption would be observed at some potential in the D mode, but since the Q mode would always begin each pulse in the same state it would give no indication of the desorption. Only in the derivative mode (D), is it necessary to give the value of the pulse height in addition to the other parameters required for all modes. Delay time is the time between pulses. If a dropping mercury electrode were to be used, the drop would be dislodged at the end of each pulse prior to the delay time.
Pulse Polarography. An approximate theory of differential pulse polarography is developed here for stationary electrodes. For details, see Appendix 11. The results are essentially the same as those derived for derivative pulse polarography at a dropping electrode, and the reader is referred to the original papersfor details (16-19). The approximate model used here is the application of a potential step from a region of no Faradaic reaction to the potential of interest followed at a time T by a pulse of height AE and duration t. The concentration gradient near the electrode is assumed to be linear when necessary. This assumption should be good for
344
7
>> t .
For “totally reversible” systems, the concentrations of reactant, C , , and product, C,, at the electrode surface are determined by the electrode potential, the bulk concentrations, and the diffusion coefficients, D, and D,. They are time invariant at any fixed potential. When the Fick’s law equations are solved, the concentration gradient prior to pulse application is found to be missing from the expression for Ai. 1-6
nFADh12CTyo Ai =
,1/zt1/2
(By”
+ 6)(Y0 + 6 )
(1)
See appendix I for notation. Note that this expression also does not depend on T and, further, that it is symmetrical about Ell2 ‘/&E, where El/z is the polarographic half-wave potential. This is essentially the same expression as those derived previously (16,18). Derivation of the equivalent equation for a stationary sphericai electrode yields the standard spherical correction term for potentiostatic processes (20). A “totally irreversible” process (see Appendix 11) results in an equation containing r.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
+
Ai = nFADrl”Tro(AO- A) exp (Ao%)
erfc (
X
exp (h2t) erfc ( ~ t ’ l z ) (2)
~ ~ ~ 1 ’ 2 )
(16) G. C. Barker and A. W. Gardner, At. Energy Res. Estab. Harwell, C/R 2297 (1958). (17) G . C . Barker, R. L. Faircloth, and A. W. Gardner, ibid.,C/R 1786. (18) E. P. Parry and R. A. Osteryoung, ANAL. CHEM., 37, 1634 (1965). (19) A. A. M. Brinkman and J. M. Los, J . Electroanal. Chem., 7,171 (1964). (20) W. H. Reinmuth, J . Amer. Chem. SOC.,79,6358 (1957).
Table I. Concentration of Cd(I1) us. Peak Height Peak height Concentration (@A) (M) 4
4 4 4
x x
x x
10-6 10-6
10-7 10-8
.
51.
I .4
* * . . a
..* *. ....*... * ..
... . . . a
0.92 0.31
This equation becomes identical to that given by Barker (16) if AE is small and the XO2r terms are combined with C,". The dependence of 7 shows that for long experiments-i.e., the effective value of T is large-sensitivity to irreversible reactions is decreased. Barker presents the same argument (16). Note that the concentration gradient again does not appear in the expression. However, higher than firstorder terms would appear if included in the derivation. Their inclusion would, however, make the mathematics unduly complex. Therefore, it is important when doing analysis by derivative pulse polarography to try to have the sought after species in a form which displays reasonable electrochemical reversibility at times of the order of the pulse width. A significant corollary to this behavior of irreversible systems is a decrease in the sensitivity to oxygen so that it will cause less interference than in, say, normal (integral mode) pulse polarography. Signal Averaging. The theory of cross correlation is well developed (7, 21). The signal is multiplied by a function synchronized with the signal. The signal is assumed to be periodic and its product with the cross correlation function is summed over the number of periods or cycles run to obtain a single output value. Several such summations may be seen simultaneously or seriatim to provide an ensemble average. In this work the cross correlation function is a periodic binary sampling function synchronized with the beginning of each pulse. In effect, the result is ensemble averaging. It is reasonable to expect that for most random noise distributions encountered in practice, the signal-to-noise ratio will increase as the square root of the averaging cycles. Digital Least Squares Smoothing. The method of smoothing employed for this paper is that given by Savitsky and Golay (22). The reader is referred to their paper and references for further details. RESULTS AND DISCUSSION
Pulse polarograms were run on cadmium nitrate in potassium nitrate supporting electrolyte with the computerized pulse polarography system. The concentrations of cadmium ranged from 4 X 10-5M to 4 X 10-8M while the corresponding K N 0 3 concentrations were 0.5M to 0.5mM. Peak height as a function of concentration is given in Table I. There is an apparent increase of sensitivity at lower concentrations. While the dilutions were made with distilled water, thus lowering the ionic strength, resultant double layer effects or migration effects would be expected to be minor. The more likely cause of the loss of linearity is an instrumental one produced by increasing solution and current measuring resistances. With increasing uncompensated resistance, the potentiostat does not control until (21) D. J. Fisher, Chem. Instrum., 2, 1 (1969). (22) A. Savitsky and M. J. E. Golay, ANAL.CHEM., 36,1627 (1964).
I
-.4
-.6 -.8 VOLTS VS SCE
Figure 3. Derivative pulse polarogram of 4 X lO-*M Cd*+, 5 X 10-4MKN03 20-cycle average without smoothing. A = 0.0414 cm2. Pulse height = 50 mV. Pulse width = 20 msec. Delay = 200msec
PULSE WIDTH
V
t
Figure 4. Voltage wave form for derivative mode pulse polarography several milliseconds after the pulse is applied. The peak heights are readily reproducible at all except the lowest concentration where extreme care must be taken to eliminate oxygen. Figure 3 shows the response of the system to the lowest concentration studied, 4 X 10-8M Cd2+ in 0.5 X 10-3M KNOs. A readily measurable peak is apparent. At higher concentrations, a large, well-formed peak is observed. With proper precautions, the detection limit for computerized pulse polarography of cadmium should be substantially less than 10-8M. This is better than has been previously reported for pulse polarography. The reproducibility of the system described in this paper is demonstrated by a series of replicate runs. Data were output on the teletype. The largest value was taken as the peak, while the minimum value preceding the peak was taken as the base line. Four runs at 4 X 10-6M Cd2+ had a standard deviation of 1.5% while ten at 4 X 10d7MCd2+ had a standard deviation of 10%. The drop area was reproducible to only about 1% accounting for most of the error in the first series. Since the background in the second series had a magnitude of roughly one-half of the peak and was not necessarily additive, it is likely that this is the predominant source of error. A longer pulse width than the 20 milliseconds employed plus better electronic circuitry should be sufficient to reduce the error considerably. Better oxygen scrubbing techniques are also indicated.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
345
1
1
.-160mv
. . :-140mv
,
6 0 m i -.:
- .2
. ,80mv
-.2
-.6
VOLTS VS SCE
-
.6 VOLTS VS SCE
Figure 5. Derivative pulse polarograms of 4 X 10-5M Cd2+' 0.5M KNOa
Figure 7. Derivative pulse polarogram of 4 X 10-6M Cd2+, 0.05MKNO3
= 0.0414 cm2. Pulse height = 20, 40, 60, 80, 100, 120, 140, 160 mV. Pulse width = 20 msec. Delay = 200 msec
5-cycle average without smoothing. A = 0.0414 em2. Pulse height 50 mV. Pulse width = 10 msec. Delay = 100,200msec
5-cycle average without smoothing. A
, ,,,
,
50ma'
-.2
,..:....
::'..I
-.6 VOLTSVS SCE
Figure 6 . Derivative pulse polarogram of 4 X 10-6M Cd2+ 0.05MKNOa 5-cycle average without smoothing. A = 0.0414 cm2. Pulse height = 50 mV. Pulse width, delay = 10,100 msec; 20,200 msec; 50,500 msec Effort directed toward optimization of experimental parameters has been reported (3, 18). Because there are new factors present in the hanging drop method and because of the importance of optimization to achieving maximum sensitivity, certain of these parameters have been investigated here. The parameters associated with the voltage wave form are shown in Figure 4. Calculations were made on the theoretical response for variations of pulse height. There can be no general optimal value since one individual may be concerned with peak separation, while another may desire only a maximum height to half-width ratio. The effect of pulse height on separation has already been discussed elsewhere (3). The result is that smaller pulse heights provide greater separation at the expense of sensitivity. Since large pulse height will result in the peak approaching a maximum height while continually broadening and small pulses cause continually diminishing peak height but a constant, minimum peak width, it would seem that some intermediate pulse height should be optimal. An arbitrary 346
=
measure of the optimal value is the ratio of peak height to the half-peak width. Hand calculations show that this quantity is a maximum when nFAE/RT = 2.1 or AE = 63 mV for a two-electron process at 25 OC. However, equally significant is the fact that the range of pulse heights allowed if a 10% variation of the above ratio is permitted, is 180/n mV to 84/n mV at 25 "C. It is therefore quite likely that the pulse height for any given experiment will be determined by other factors. This is especially true if two species undergoing reductions requiring different numbers of electrons are being studied simultaneously or if peak separation is a problem. The actual results of variation of pulse height are shown in Figure 5. The 60-mV curve appears to be the best one. The asymmetry observed at large pulse heights is likely due to insufficient delay times between pulses. At pulse heights of 100 mV and greater, the trailing edge of the peak moves cathodic. However, if there are no nearby interfering ions, then high sensitivity can be achieved by using relatively large pulse heights. For maximum response, the pulse width should be a minimum, although if too short, double layer relaxation will make a significant contribution and limit the sensitivity. The effect of pulse widths of 10, 20, and 50 milliseconds is displayed in Figure 6. The delay time is ten times the pulse width in each case. The heights obey the t - l I 2 law predicted by theory to better than 5%. Other experiments show that if the pulse width is greater than 0.1 sec, positive deviations are observed. These deviations are readily explained by convection and spherical effects. Much more significant is the effect on the background. The extreme change in the background level observed in Figure 6 is due to instrumental limitations. Positive feedback should be capable of reducing this effect, thus allowing the instrument to achieve still greater sensitivity. The delay time between pulses allows for the relaxation of the disturbance. If the delay is too short, then the signal becomes noisy and sensitivity is decreased. Long delays increase measurement time. Therefore the delay must be a compromise between these effects. For the 50-mV pulses used in most of the work, a delay of ten times the pulse width was found to be satisfactory. Figure 7 shows the effect of 100- and 200-msec delays when the pulse width is
ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
The direct connection between the experiment and the computer with the experimentalist present gives a rapid turn-around capability which is highly beneficial in experiments of the type demonstrated. First, there is immediate knowledge of the success or failure of the experimental arrangement and parameters used. Second, quantitative results are available at the moment. No delays due to data conversion and processing are incurred. Third, there are advantages unrealized in this work such as removing the experimenter from the computer-experiment feedback loop by programming decision-making capabilities into the computer. This could even be done by computerized learning machine methodology. A program listing or source tape may be had by writing to the authors.
b
,.,; -.2
.. :
.',. :.. .'. '.
; ., ., :... .... .,. ..... . ......
-.6
-.2
-6
ACKNOWLEDGMENT
VOLTS VS SCE
Figure 8. Derivative pulse polarograms of 4 X 10-jM Cd2+, 0 S M KN03 Pulse height = 50 mV. Pulse width = 20 msec. Delay = 200 msec. (a) No smoothing, average X 1 (b) No smoothing, average X 9 (c) 9-pt smooth, average X 1 (d) 9-pt smooth, average X 9
Some of the programming ideas are due to George Lauer. The hardware interface was built by P. R. Mohilner and was based on a design for a similar interface by G . Lauer and R. A. Osteryoung. Helpful discussions with J. H. Christie are acknowledged. APPENDIX I
10 msec. A slight sensitivity improvement is observed with the 200-msec. delay. One of the primary advantages of a computerized experimental system is the acquisition and storage of data in digital form. These data can then be operated upon during or after the experiment to improve the quality of experimental results. In the present system, two means of operating on the data are employed. The data are averaged by taking several measurements at the same potential repetitively, and the data are smoothed by a least squares procedure after the experiment. Both of the operations are optional. The first has the effect of increasing the signal-to-noise ratio as the square root of the number of averaging cycles. There is no distortion created if the delay between pulses is sufficiently long to allow virtually complete relaxation of the disturbance created by the pulses. The second method improves the signal-to-noise ratio as the square root of the number of values over which each point is calculated. Distortion will be introduced if the order of the polynomial used to fit the data is too small to adequately represent the segment of the curve being fit. Test computations indicate that a nine-point quartic smooth will introduce no measurable distortion with data points spaced 10 mV apart. The two above methods affect the signal-to-noise ratio independently so that the addition of the nine point smoothing algorithm should have the same effect on the signal-to-noise ratio as multiplying the number of averaging cycles by nine. Figure 8 demonstrates the efficacy of these noise reduction techniques. In Figure Ea a raw curve is shown. Figure Eb shows the effect of averaging over nine cycles while, in Figure 8c, the nine-point smoothing routine is used. The two curves show roughly the same reduction in noise. Figure 8d is the result of applying both techniques. The effect of noise is virtually eliminated. The experimental results indicate that through the use of an on-line computer a significant increase in the sensitivity of pulse polarography can be attained. Particularly important to this increase are the techniques of signal averaging and data smoothing and the ability to vary experimental parameters readily over a wide range.
Notation
number of electrons transferred per mole of reactant Faraday electrode area diffusion coefficient of reactant diffusion coefficient of product concentrations concentrations at t = 0, x = 0 D,l! ?/D,l'2
+
C," 6CPZ exp (rzF/RT L E ) pulse height exp [riF/RT(E,- E")] potential prior to application of pulse
formal standard potential of electrode reaction k,D,-' exp [-curzF/RT(E, - E")] A, exp (- ariF/RTAE)
formal standard heterogeneous rate constant 1 - erf ( x ) where erf(x) = 2/;r1/2J0z e-t2df current current at t = 0 k , exp [arzF/RT(EI- E")] i D,-"* [/iF/RT(E,- E " ) ] ) A( 1 6 - l exp [nF/RT(E - E ' ) ] }
+ D,-1'2 exp
+
APPENDIX I1 Derivation of Curve Shape for Derivative Mode Pulse Polarography at a Stationary Electrode
From Delahay ( 2 3 ,
i
=
nFACr0X~Dr1/2t?zr erfc ( Q T ~ z,
('41)
From Fick's diffusion equation, after Laplace transformation and using semi-infinite linear diffusion,
Cr
=
c?o/~ + 6 exp [ - x ~ ' ~ / 0 7 1
('42)
The new variable, ii, is an undetermined coefficient. Therefore, (Cr1z-o
=
CrO
+u
('43)
(23) P. Delahay, "New Instrumental Methods in Electrochemistry," Interscience, New York, N. Y . , 1954, p 74. ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
347
and (ac7/ax)z=o=
-ads,
(-441
Substitute Equations A13, A15, and A19 into A17 and A14 into A18; equate currents to get
However, using Equation A l ,
Combining Equations A4 and A5 yields
nFAD,(cl/s - 6 1 / s / D r ) (A20)
Solve for 6 using (see Appendix I),
a Substituting A6 into A3 and A4 and inverting the transform gives erfc (Q7’”)I Cr(0,T) c? - C70XoQ-1[1 - eQZr (-47)
c1D7”2 i ; = s(s”2 p)
+
x
S(S”2
+ p) X
and Using Equations A18, A14, and A21 and inverting the transform
Similarly, Cp(0,7)= C,oXo6-1Q-1[l - eQzrerfc ( Q T ” ~ ) ]
(A9)
c1
- clefiZterfc ( ~ t l ! ~+)
and
Now, assume that the concentration gradients are linear and can be represented by
c, = co + CIX and
c,
= co’
+
CI‘X
(A1 la)
Replacing CO,co’, cl, and cl’ with the values given by Equations A7, A9, AS, and A10 and subtracting A1 gives
(Allb)
Once again, solving Fick‘s equation in Laplace space yields
C,
1 = ;(co
+ clx) + 6 exp [- x d = ]
(1
1
- eQZrerfc (Q71’2))eP’terfc (jL,t1I2)
(AW
-
XoeQZr erfc (QT1/2)efi*t erfc (jL,t1/2)}
(A23)
So that C,(O,s)
= cojs
+6
(~13)
and der(0,S) --a x
-
Cl/S
-6 6 ,
Equation A23 is the general expression for a pulse applied at time 7 after a step, using the approximation of linear concentration gradients. The limiting case of a totally irreversible reaction may be obtained from the following limits:
[
Q -c io, p + A, exp g ( E - Eo)] -.* 0.
Similarly, Cp(0,S) =
cor
+ 6’
(A1 5)
The result is Ai
and
=
nFADf1/2Cfo(X - XO)eXazr erfc (XoT1/2)eX2terfc (Xt1’2) (‘424)
Substituting into the absolute rate theory expression (24),
The limiting case of a totally reversible reaction can be obtained by taking k, -P a . Then p + m , X + m and
The result is
Ai = Note that i = nFAD,(bC,/bx)z,o
(‘418)
Assume equal and opposite fluxes (23) and use Equations A14 and A16 with appropriate manipulation to give
6’
=
~ , ’ D P l / 2 ~ - 3+/ 2clD,1/2~-ls-312 - 66-1
(A19)
(24) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, N. Y.,1954, p 34. 348
ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
nFACfO D P 12yo / rl/zt’/z
1 - P
(rO+ W P Y 0 + 8 )
(A25)
In the above derivation, an electrochemical reduction has been assumed and the initial concentration of the product was zero. In the reversible case, the initial concentration of the product may be non-zero and the requirement of linearity of the concentration gradient may be relaxed. RECEIVED for review August 13, 1970. Accepted December 21, 1970. This work was supported in part by NASA Grant Number NGR-06-002-088.
